The lifespans of snakes in a particular zoo are normally distributed. The average snake lives $22$ years; the standard deviation is $4.6$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a snake living between $31.2$ and $35.8$ years.
Answer: $22$ $17.4$ $26.6$ $12.8$ $31.2$ $8.2$ $35.8$ $99.7\%$ $95\%$ $2.35\%$ $2.35\%$ We know the lifespans are normally distributed with an average lifespan of $22$ years. We know the standard deviation is $4.6$ years, so one standard deviation below the mean is $17.4$ years and one standard deviation above the mean is $26.6$ years. Two standard deviations below the mean is $12.8$ years and two standard deviations above the mean is $31.2$ years. Three standard deviations below the mean is $8.2$ years and three standard deviations above the mean is $35.8$ years. We are interested in the probability of a snake living between $31.2$ and $35.8$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the snakes will have lifespans within 3 standard deviations of the average lifespan. It also tells us that $95\%$ of the snakes will have lifespans within 2 standard deviations of the mean. That leaves $99.7\% - 95\% = 4.7\%$ of snakes between 2 and 3 standard deviations of the mean, or $2.35\%$ on either side of the distribution. The probability of a particular snake living between $31.2$ and $35.8$ years is $\color{orange}{2.35\%}$.